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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1uplex | Structured version Visualization version GIF version | ||
| Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.) | 
| Ref | Expression | 
|---|---|
| bj-1uplex | ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-pr11val 37007 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
| 2 | bj-pr1ex 37008 | . . 3 ⊢ (⦅𝐴⦆ ∈ V → pr1 ⦅𝐴⦆ ∈ V) | |
| 3 | 1, 2 | eqeltrrid 2845 | . 2 ⊢ (⦅𝐴⦆ ∈ V → 𝐴 ∈ V) | 
| 4 | df-bj-1upl 37000 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
| 5 | p0ex 5383 | . . . 4 ⊢ {∅} ∈ V | |
| 6 | bj-xtagex 36991 | . . . 4 ⊢ ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)) | |
| 7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V) | 
| 8 | 4, 7 | eqeltrid 2844 | . 2 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) | 
| 9 | 3, 8 | impbii 209 | 1 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 {csn 4625 × cxp 5682 tag bj-ctag 36976 ⦅bj-c1upl 36999 pr1 bj-cpr1 37002 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-bj-sngl 36968 df-bj-tag 36977 df-bj-proj 36993 df-bj-1upl 37000 df-bj-pr1 37003 | 
| This theorem is referenced by: bj-2uplex 37024 | 
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